ZLAED8(3)      LAPACK routine of NEC Numeric Library Collection      ZLAED8(3)



NAME
       ZLAED8

SYNOPSIS
       SUBROUTINE ZLAED8 (K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2,
           LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO)



PURPOSE
            ZLAED8 merges the two sets of eigenvalues together into a single
            sorted set.  Then it tries to deflate the size of the problem.
            There are two ways in which deflation can occur:  when two or more
            eigenvalues are close together or if there is a tiny element in the
            Z vector.  For each such occurrence the order of the related secular
            equation problem is reduced by one.




ARGUMENTS
           K         (output)
                     K is INTEGER
                    Contains the number of non-deflated eigenvalues.
                    This is the order of the related secular equation.

           N         (input)
                     N is INTEGER
                    The dimension of the symmetric tridiagonal matrix.  N >= 0.

           QSIZ      (input)
                     QSIZ is INTEGER
                    The dimension of the unitary matrix used to reduce
                    the dense or band matrix to tridiagonal form.
                    QSIZ >= N if ICOMPQ = 1.

           Q         (input/output)
                     Q is COMPLEX*16 array, dimension (LDQ,N)
                    On entry, Q contains the eigenvectors of the partially solved
                    system which has been previously updated in matrix
                    multiplies with other partially solved eigensystems.
                    On exit, Q contains the trailing (N-K) updated eigenvectors
                    (those which were deflated) in its last N-K columns.

           LDQ       (input)
                     LDQ is INTEGER
                    The leading dimension of the array Q.  LDQ >= max( 1, N ).

           D         (input/output)
                     D is DOUBLE PRECISION array, dimension (N)
                    On entry, D contains the eigenvalues of the two submatrices to
                    be combined.  On exit, D contains the trailing (N-K) updated
                    eigenvalues (those which were deflated) sorted into increasing
                    order.

           RHO       (input/output)
                     RHO is DOUBLE PRECISION
                    Contains the off diagonal element associated with the rank-1
                    cut which originally split the two submatrices which are now
                    being recombined. RHO is modified during the computation to
                    the value required by DLAED3.

           CUTPNT    (input)
                     CUTPNT is INTEGER
                    Contains the location of the last eigenvalue in the leading
                    sub-matrix.  MIN(1,N) <= CUTPNT <= N.

           Z         (input)
                     Z is DOUBLE PRECISION array, dimension (N)
                    On input this vector contains the updating vector (the last
                    row of the first sub-eigenvector matrix and the first row of
                    the second sub-eigenvector matrix).  The contents of Z are
                    destroyed during the updating process.

           DLAMDA    (output)
                     DLAMDA is DOUBLE PRECISION array, dimension (N)
                    Contains a copy of the first K eigenvalues which will be used
                    by DLAED3 to form the secular equation.

           Q2        (output)
                     Q2 is COMPLEX*16 array, dimension (LDQ2,N)
                    If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
                    Contains a copy of the first K eigenvectors which will be used
                    by DLAED7 in a matrix multiply (DGEMM) to update the new
                    eigenvectors.

           LDQ2      (input)
                     LDQ2 is INTEGER
                    The leading dimension of the array Q2.  LDQ2 >= max( 1, N ).

           W         (output)
                     W is DOUBLE PRECISION array, dimension (N)
                    This will hold the first k values of the final
                    deflation-altered z-vector and will be passed to DLAED3.

           INDXP     (output)
                     INDXP is INTEGER array, dimension (N)
                    This will contain the permutation used to place deflated
                    values of D at the end of the array. On output INDXP(1:K)
                    points to the nondeflated D-values and INDXP(K+1:N)
                    points to the deflated eigenvalues.

           INDX      (output)
                     INDX is INTEGER array, dimension (N)
                    This will contain the permutation used to sort the contents of
                    D into ascending order.

           INDXQ     (input)
                     INDXQ is INTEGER array, dimension (N)
                    This contains the permutation which separately sorts the two
                    sub-problems in D into ascending order.  Note that elements in
                    the second half of this permutation must first have CUTPNT
                    added to their values in order to be accurate.

           PERM      (output)
                     PERM is INTEGER array, dimension (N)
                    Contains the permutations (from deflation and sorting) to be
                    applied to each eigenblock.

           GIVPTR    (output)
                     GIVPTR is INTEGER
                    Contains the number of Givens rotations which took place in
                    this subproblem.

           GIVCOL    (output)
                     GIVCOL is INTEGER array, dimension (2, N)
                    Each pair of numbers indicates a pair of columns to take place
                    in a Givens rotation.

           GIVNUM    (output)
                     GIVNUM is DOUBLE PRECISION array, dimension (2, N)
                    Each number indicates the S value to be used in the
                    corresponding Givens rotation.

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.



LAPACK routine                  31 October 2017                      ZLAED8(3)